Some Graph-Theoretical Aspects of the Golden Ratio: Topological Index, Isomatching Graphs, and Golden Family Graphs

Haruo Hosoya

Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, Japan
E-mail address: hosoya@is.ocha.ac.jp

(Received February 15, 2005; Accepted March 15, 2005)

Keywords: Fibonacci Numbers, Lucas Numbers, Golden Family Graph, Topological Index, Non-adjacent number

Abstract. By defining the non-adjacent number, p(G, k), and topological index, Z_G, for a graph G, several sequences of graphs are shown to be closely related to the golden ratio, t. Namely, the Z-values of the path and cycle graphs are Fibonacci, and Lucas numbers, respectively, and thus the ratio of consecutive terms of Z converges to t. Several new sequences of graphs (golden family graphs) were found whose Z-values are either Fibonacci or Lucas numbers, or their multiples. Interesting mathematical relations among them are introduced and discussed.